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A002415 4-dimensional pyramidal numbers: n^2*(n^2-1)/12.
(Formerly M4135 N1714)
82
0, 0, 1, 6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, 19481, 23276, 27600, 32500, 38025, 44226, 51156, 58870, 67425, 76880, 87296, 98736, 111265, 124950, 139860, 156066, 173641, 192660, 213200, 235340 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Also number of ways to legally insert two pairs of parentheses into a string of m := n-1 letters. (There are initially 2C(m+4,4) (A034827) ways to insert the parentheses, but we must subtract 2(m+1) for illegal clumps of 4 parentheses, 2m(m+1) for clumps of 3 parentheses, C(m+1,2) for 2 clumps of 2 parentheses and (m-1)C(m+1,2) for 1 clump of 2 parentheses, giving m(m+1)^2(m+2)/12 = n^2*(n^2-1)/12.) See also A000217.

E.g., for n=2 there are 6 ways: ((a))b, ((a)b), ((ab)), (a)(b), (a(b)), a((b)).

Let M_n denote the n X n matrix M_n(i,j)=(i+j); then the characteristic polynomial of M_n is x^(n-2) * (x^2-A002378(n)*x - a(n)). - Benoit Cloitre, Nov 09 2002

Let M_n denote the n X n matrix M_n(i,j)=(i-j); then the characteristic polynomial of M_n is x^n + a(n)x^(n-2). - Michael Somos, Nov 14 2002

a(n)+1 is the determinant of the n X n matrix M with M(i,i)=1, M(i,j)=i-j. - Mario Catalani (mario.catalani(AT)unito.it), Feb 12 2003

Number of permutations of [n] which avoid the pattern 132 and have exactly 2 descents. - Mike Zabrocki, Aug 26 2004

Number of tilings of a <2,n,2> hexagon.

a(n) = number of squares with corners on an n X n grid. See also A024206, A108279.

Kekule numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005

Number of distinct components of the Riemann curvature tensor. - Gene Ward Smith, Apr 24 2006

Convolution of natural numbers (A001477) with squares (A000290). - Graeme McRae, Jun 06 2006

a(n) is the number of 4 X 4 matrices (symmetrical about each diagonal) M = [a,b,c,d;b,e,f,c;c,f,e,b;d,c,b,a] with a+b+c+d=b+e+f+c=n+2; (a,b,c,d,e,f natural numbers). - Philippe Deléham, Apr 11 2007

If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-3) is the number of 5-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007

a(n) = number of Dyck (n+1)-paths with exactly n-1 peaks. - David Callan, Sep 20 2007

Starting (1,6,20,50,...) = third partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = sum{i=0,n,C(n+3,i+3)*b(i)}, where b(i)=[1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009

4-dimensional square numbers. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009

Equals row sums of triangle A177877; a(n), n>1 = (n-1) terms in (1,2,3,...) dot (...3,2,1) with additive carryovers. Example: a(4) = 20 = (1,2,3) dot (3,2,1) with carryovers = (1*3) + (2*2 + 3) + (3*1 + 7) = (3 + 7 + 10).

Convolution of the triangular numbers A000217 with the odd numbers A004273.

a(n+2) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and w-x=max{w,x,y,z}-min{w,x,y,z}. - Clark Kimberling, May 28 2012

The second level of finite differences is a(n+2)-2*a(n+1)+a(n) = (n+1)^2, the squares. - J. M. Bergot, May 29 2012

Because the differences of this sequence give A000330, this is also the number of squares in an n+1 X n+1 grid whose sides are not parallel to the axes.

a(n+2) gives the number of 2*2 arrays that can be populated with 0..n such that rows and columns are nondecreasing. - Jon Perry, Mar 30 2013

For n consecutive numbers 1,2,3,....n, the sum of all ways of adding the k-tuples of consecutive numbers for n=a(n+1). As an example, let n=4: (1)+(2)+(3)+(4)=10; (1+2)+(2+3)+(3+4)=15; (1+2+3)+(2+3+4)=15; (1+2+3+4)=10 and the sum of these is 50=a(4+1)=a(5). - J. M. Bergot, Apr 19 2013

If P(n,k) = n*(n+1)*(k*n-k+3)/6 is the n-th (k+2)-gonal pyramidal number, then a(n) = P(n,k)*P(n-1,k-1) - P(n-1,k)*P(n,k-1). - Bruno Berselli, Feb 18 2014

For n>1, a(n) = 1/6 of the area of the trapezoid created by the points (n,n+1), (n+1,n), (1,n^2+n), (n^2+n,1). - J. M. Bergot, May 14 2014

For n>3, a(n) is twice the area of a triangle with vertices at points

(C(n,4),C(n+1,4)), (C(n+1,4),C(n+2,4)), and (C(n+2,4),C(n+3,4)). - J. M. Bergot, Jun 03 2014

REFERENCES

O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.

S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165).

R. Euler and J. Sadek, "The Number of Squares on a Geoboard", Journal of Recreational Mathematics, 251-5 30(4) 1999-2000 Baywood Pub. NY

G. Kreweras, Traitement simultane du "Probleme de Young" et du "Probleme de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle. Institut de Statistique, Universit\'{e} de Paris, 10 (1967), 23-31.

S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 238.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

H. Bottomley, Illustration of initial terms

Duane DeTemple, Using Squares to Sum Squares, The College Mathematics Journal, ? (2010), 214-221. [From Parthasarathy Nambi, Jul 16 2010]

Reinhard O. W. Franz, and Berton A. Earnshaw, A constructive enumeration of meanders, Ann. Comb. 6 (2002), no. 1, 7-17.

M. Hyatt and J. Remmel, The classification of 231-avoiding permutations by descents and maximum drop, Arxiv preprint arXiv:1208.1052, 2012. - From N. J. A. Sloane, Dec 24 2012

Milan Janjic, Two Enumerative Functions

M. Jones, S. Kitaev, J. Remmel, Frame patterns in n-cycles, arXiv preprint arXiv:1311.3332, 2013

C. P. Neuman, D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698).

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

P. N. Rathie, A census of simple planar triangulations, J. Combin. Theory, B 16 (1974), 134-138. See Table I. - N. J. A. Sloane, Mar 27 2012

Royce A. Speck, The Number of Squares on a Geoboard, School Science and Mathematics, Volume 79, Issue 2, pages 145-150, February 1979

Eric Weisstein's World of Mathematics, Riemann Tensor.

Index entries for sequences related to Chebyshev polynomials.

Index to sequences with linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

G.f.: x^2*(1+x)/(1-x)^5. - Simon Plouffe in his 1992 dissertation

a(n) = sum(i=0..n, (n-i)*i^2) = a(n-1)+A000330(n-1) = A000217(n)*A000292(n-2)/n = A000217(n)*A000217(n-1)/3 = A006011(n-1)/3. - Henry Bottomley, Oct 19 2000

a(n) = 2*C(n+2, 4) - C(n+1, 3). - Paul Barry, Mar 04 2003

a(n) = C(n+2, 4) + C(n+3, 4). - Paul Barry, Mar 13 2003

a(n) = sum(k=1, n, A000330(n-1)). - Benoit Cloitre, Jun 15 2003

a(n) = n*C(n+1,3)/2 = C(n+1,3)*C(n+1,2)/(n+1). - Mitch Harris, Jul 06 2006

a(n) = A006011(n)/3 = A008911(n)/2 = A047928(n-1)/12 = A083374(n)/6. - Zerinvary Lajos, May 09 2007

a(n) = 1/2*sum {1 <= x_1, x_2 <= n} (det V(x_1,x_2))^2 = 1/2*sum {1 <= i,j <= n} (i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007

a(n) = C(n+1,3) + 2*C(n+1,4). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009

a(n) = (1/48)sinh(2*arccosh(n))^2. - Artur Jasinski, Feb 10 2010

a(n) = n*A000292(n-1)/2. - Tom Copeland, Sep 13 2011

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5), n>4. - Harvey P. Dale, Nov 29 2011

a(n) = (n-1)*A000217(n-1) - sum((n-1-2*i)*A000217(i), i=0..n-2) for n>1. - Bruno Berselli, Jun 22 2013

a(n) = C(n,2)*C(n+1,3) - C(n,3)*C(n+1,2). - J. M. Bergot, Sep 17 2013

a(n) = sum_{k=1..n} ( (2k-n)* k(k+1)/2 ). - Wesley Ivan Hurt, Sep 26 2013

a(n) = floor(n^2/3) + 3*sum{k=1..n,k^2*floor((n-k+1)/3)}. - Mircea Merca, Feb 06 2014

Euler transform of length 2 sequence [ 6, -1]. - Michael Somos, May 28 2014

a(n) = sum_{i=1..n} i*(n-i)^2. - Wesley Ivan Hurt, Aug 21 2014

EXAMPLE

a(7) = 6*21 - (6*0+4*1+2*3+0*6-2*10-4*15) = 196. - Bruno Berselli, Jun 22 2013

G.f. = x^2 + 6*x^3 + 20*x^4 + 50*x^5 + 105*x^6 + 196*x^7 + 336*x^8 + ...

MAPLE

A002415 := proc(n) binomial(n^2, 2)/6 ; end proc: # Zerinvary Lajos, Jan 07 2008

MATHEMATICA

Table[(n^4 - n^2)/12, {n, 0, 40}] (* Zerinvary Lajos, Mar 21 2007 *)

Table[(1/48) Round[N[Sinh[2 ArcCosh[n]]^2, 100]], {n, 0, 50}] (* Artur Jasinski, Feb 10 2010 *)

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 1, 6, 20}, 40] (* Harvey P. Dale, Nov 29 2011 *)

PROG

(PARI) {a(n) = n^2 * (n^2 - 1) / 12};

(PARI) a(n)=sum(k=1, n, sum(m=1, k, sum(i=1, m, (2*i-1)))); \\ Alexander R. Povolotsky, Nov 05 2007

(MAGMA) [n^2*(n^2-1)/12: n in [0..50]]; // Wesley Ivan Hurt, May 14 2014

CROSSREFS

a(n) = ((-1)^n)*A053120(2*n, 4)/8 (one eighth of fifth unsigned column of Chebyshev T-triangle, zeros omitted). Cf. A001296.

Second row of array A103905.

Third column of Narayana numbers A001263.

Cf. A001079, A002415, A006011, A006542, A008911, A037270, A047819, A047928, A071253, A083374, A107891, A108741, A132592, A146311, A146312, A146313, A173115, A173116.

Partial sums of A000330.

The expression binomial(m+n-1,n)^2-binomial(m+n,n+1)*binomial(m+n-2,n-1) for the values m = 2 through 14 produces sequences A000012, A000217, A002415, A006542, A006857, A108679, A134288, A134289, A134290, A134291, A140925, A140935, A169937.

Cf. A220212 for a list of sequences produced by the convolution of the natural numbers (A000027) with the k-gonal numbers.

Sequence in context: A161699 A216175 A161409 * A052515 A067117 A213586

Adjacent sequences:  A002412 A002413 A002414 * A002416 A002417 A002418

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Apr 30 1991

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Oct 19 2000

Typo in link fixed by Matthew Vandermast, Nov 22 2010

Redundant comment deleted and more detail on relationship with A000330 added by Joshua Zucker, Jan 01 2013

STATUS

approved

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Last modified September 16 11:41 EDT 2014. Contains 246817 sequences.